3.505 \(\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=61 \[ -\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d} \]

[Out]

-(a*Cot[c + d*x]^6)/(6*d) - (a*Csc[c + d*x])/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.103967, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2607, 30, 2606, 194} \[ -\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^6)/(6*d) - (a*Csc[c + d*x])/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)

Rule 2834

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc (c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc (c+d x)}{d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0242166, size = 61, normalized size = 1. \[ -\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^6)/(6*d) - (a*Csc[c + d*x])/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)

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Maple [A]  time = 0.061, size = 110, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{15\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/5/sin(d*x+c)^5*cos(d*x+c)^6+1/15/sin(d*x+c)^3*cos(d*x+c)^6-1/5/sin(d*x+c)*cos(d*x+c)^6-1/5*(8/3+cos
(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-1/6*a/sin(d*x+c)^6*cos(d*x+c)^6)

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Maxima [A]  time = 1.13625, size = 95, normalized size = 1.56 \begin{align*} -\frac{30 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, a \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/30*(30*a*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 20*a*sin(d*x + c)^3 - 15*a*sin(d*x + c)^2 + 6*a*sin(d*x + c
) + 5*a)/(d*sin(d*x + c)^6)

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Fricas [A]  time = 1.06001, size = 254, normalized size = 4.16 \begin{align*} \frac{15 \, a \cos \left (d x + c\right )^{4} - 15 \, a \cos \left (d x + c\right )^{2} + 2 \,{\left (15 \, a \cos \left (d x + c\right )^{4} - 20 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 5 \, a}{30 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/30*(15*a*cos(d*x + c)^4 - 15*a*cos(d*x + c)^2 + 2*(15*a*cos(d*x + c)^4 - 20*a*cos(d*x + c)^2 + 8*a)*sin(d*x
+ c) + 5*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**7*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.33398, size = 95, normalized size = 1.56 \begin{align*} -\frac{30 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, a \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/30*(30*a*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 20*a*sin(d*x + c)^3 - 15*a*sin(d*x + c)^2 + 6*a*sin(d*x + c
) + 5*a)/(d*sin(d*x + c)^6)